Optimal temperature sensor placement

3.1 Introduction

In many engineering applications, it is important to determine the state of a system or sub-system using only partial measurements of that system states. Of particular interest in hybrid electric vehicle applications is the measurement of the cell temperature in a battery pack. Monitoring temperature allows for control algorithms that can prevent excessive tem-perature rise and ultimately prevent thermal runways [14]. Although higher temtem-peratures enhance the lithium transport kinetics, it is well established that excessive temperatures can also result in degradation in battery cells [56,57,58], and must be monitored.

Complex and expensive packaging and engineering schemes are required to position temperature sensors close to the regions of interest. Insertion of thermistors between bat-tery cells is infeasible because the thickness of the sensors (>1mm) will cause failure or rupture of the cells due to compression on the thick sensors. However, new sensor tech-nologies are emerging that enable the placement of temperature sensors anywhere on the cell surface at lower cost and without causing risk of battery failure. These sensors are thin film RTD (resistance temperature detector) sensors that are supplied by General Electric Global Research. The RTDs have a low profile, flexible structure that allows placement between cells without obstructing airflow for cooling or possibly damaging the cells. The RTDs are fabricated on a flexible Kapton substrate and the elements are composed of Plat-inum (Pt) with a nominal 100 ohm resistance. The total thickness of the sensor is less than 100 microns which is 10x thinner than the current state of the art temperature sensors used on electric vehicles. The sensors can be placed anywhere on the surface of the battery as shown in Fig.3.1. This motivates an observability analysis to determine the optimal loca-tion for a sensor on the surface of the cell. Since the core temperatures of the cell are of interest, maximizing observability, in this chapter, is defined as identifying optimal sensor locations for the best estimation of the core temperatures.

Figure 3.1: RTD sensor placement on the surface of a battery.

Optimal sensor and actuator placement has been the focus of significant research in the controls community. The optimal locations for sensors and actuators for parabolic PDEs, which include the heat equation, have been widely investigated [59,60]. In [61], Georges looked at the optimal sensor and actuator locations in both discrete and continuous time domains based on the observability and controllability gramians. This concept of looking at observability for defining optimal sensor placements is of major importance because it allows for the best estimation of unknown system states using the sensor data. Several papers have looked at aspects of observability for PDEs [62,63]. In [64], Wolf et al. studied the optimal sensor placement in battery packs by performing eigen decomposition of the heat equation PDEs that govern the entire battery pack and looking at the magnitude of their corresponding eigenmodes. However, little work has been done on optimal temperature sensor placements on a single battery cell.

Two methods are presented here for studying the observability and optimal temperature sensor placement on a battery cell with air flow (convection) over its surface to simulate battery pack conditions. The first is an extension to the method presented by Lim where

observability is defined in terms of the projections of the eigenvectors on the observability subspace [27], while the second looks at the trace of the gramian matrix [65]. The ad-vantage of using the first method is that it is possible to maximize observability of certain nodes of interest, such as core nodes of a battery cell. The second method looks at maxi-mizing observability for the system as a whole and is a more traditional method of looking at observability.

3.2 State space Representation

The model presented in Chapter2is nonlinear. The nonlinearity is present in the form of temperature dependent resistances (Rs, R₁and R2), and entropic heat generation IT^dU_dT. For any given profile or excitation input, the steady state value is found using the non-linear model, and non-linearization is done around that steady state value for both the tempera-ture dependent resistances (R_s, R₁and R₂), and the entropic heat generation IT^dU_dT, where for the resistance:

R(T ) = R(T_eq) + ∂R

∂T    Teq

(T − T_eq). (3.1)

and for the entropic heat generation f (I², T ) = IT^dU_dT:

f (I², T ) = f (I_eq², T_eq) + ∂f

∂I²(I²− I_eq²) + ∂f

∂T(T − T_eq). (3.2) where _∂I^∂f2 is calculated as:

∂f

∂I² = 1

2I_eq(T_eqdU

dT ). (3.3)

Note that linearization is done with respect to I² and T since I² is an input to the sys-tem, and T is a state variable. Accordingly, the system can thus be written in state space representation as:

T =AT + Bu.˙

y =CT. (3.4)

Where T = [T₁, T₂, ..., T_n]^T is the matrix of state variables of temperatures at each node and u = [I², T_amb]^T is the input to the system. A and B are the system matrices that correspond to linearizing Eqs.2.2and2.9and C is a matrix defining the locations of sensors

on the system, and thus the observable output y, where:

C = h

0 1 ... 0 i

. (3.5)

indicates that, for example, only one sensor at node 2 is placed.

3.3 Sensor placement and observability

Several methods have been proposed for studying the observability of ODEs and ad-dressing the issue of optimal sensor placement. The most common metric is the condition number or minimum singular values of the observability gramian matrix W^o. However, for studying the observability of the heat equation, it has been shown that increasing the num-ber of modes of a system will rapidly decrease the value of the smallest eigenvalue, σ^N_min, of the observability gramian matrix implying weak observability [66]. After N=8 modes, the smallest eigenvalue, σ_min^N , is almost zero. For heat transfer problems, one would be interested in looking at the observability of certain modes or eigenvalues instead of all the eigenvalues of the system. Hence, a different approach for looking at observability has to be established.

Two methods for quantifying sensor placement are analyzed and compared. The first method is based on analyzing the trace of the observability gramian matrix similar to the work done by Fang et al. [65]. This method leverages the fact that a larger trace of the observability gramian matrix W^o tends to result in a higher rank for the matrix [67]. The other method is based upon the work by Lim [27], where optimal sensor placement is found using the orthogonal projections of the eigenmodes onto the observable subspace.

This method is expanded upon by studying the projections of certain eigenmodes that are of interest to the application. Thus maximizing the observability of certain nodes instead of the system as a whole.

3.3.1 Trace analysis

For the system defined in Section 3.2, and for a chosen sensor location as defined by matrix C_i, the observability gramian matrix W_i^ois defined as:

W_i^o = Z ∞

0

e^ATτC_i^TC_ie^Aτdτ. (3.6)

The trace of the matrix W_i^ois defined as:

The method presented applies [27] to achieve maximum observability for certain criti-cal nodes of interest. Those criticriti-cal nodes are the hottest nodes that correspond to the jelly roll node which cannot be measured. Given the system presented in Section 3.2, a given matrix C_i for the locations of sensors and the resulting observability gramian matrix W_i^o presented in section 3.3.1, one can decompose W_i^o into its eigen decomposition, which would be written as:

where U_i is a matrix with column vectors that form an orthogonal basis for matrix W_i⁰, and Po

i = diag(λi,1, λi,2, ..., λi,q) is a diagonal matrix of the eigenvalues of W_i^o (where λi,1 ≥ λi,2 ≥ ... ≥ λi,q), and q is the rank of the gramian matrix W_i^o or the dimension of matrix Ui (the threshold for defining the rank is set by default at 9.2 × 10⁻¹³ using MATLAB). As outlined in [27], the projection of the eigenvectors, φij, corresponding to eigenvalues λ_ij onto the subspace U_iis:

φ^proj_ij = U_i(U_i^TU_i)⁻¹U_i^Tφ_ij. (3.9) A scalar, αij, is defined to reflect the relative significance of the corresponding eigenvector φ_ij, where:

α_ij = φ^proj_ij ^TW_i^oφ^proj_ij . (3.10) Equation3.10 implies that α_ij takes a larger value when the projected eigenvector, φ^proj_ij , is in the direction of maximum observability. However, since the nodes corresponding to the jelly roll inside the cell are of interest, and observing those nodes is critical, the method presented in [27] is expanded to look at the observability of those critical nodes by analyzing the contribution of the corresponding eigenmodes of those nodes. This expansion of Lim’s method is shown in the eigenvalue plot of system matrix A in Fig.3.2. The plot in Fig. 3.2corresponds to the eigenvalues when only the chamber fan is turned on to maintain

ambient temperature (resulting in a flow rate of v = 0.65m/s over the surface of the cell).

This analysis however could be performed for different ambient and flow conditions.

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